Alice and Bob decide to share a chocolate bar, which is an $$$n$$$ by $$$m$$$ rectangular grid of chocolate cells. They decide that Alice should get $$$a \lt n \cdot m$$$ pieces and that Bob should get $$$b = n \cdot m - a$$$ pieces. To split the chocolate bar, they repeatedly take a single piece of chocolate and break it either horizontally or vertically, creating two smaller pieces of chocolate. See Figure 1 for an example.

What is the minimum number of splits that Alice and Bob need to perform in order to split the $$$n$$$-by-$$$m$$$ chocolate bar into two piles consisting of $$$a$$$ and $$$b$$$ chocolate cells?

Figure 1: Illustration of a solution to Sample Input 2, showing the original $$$10$$$-by-$$$10$$$ chocolate bar split three times into pieces of size $$$10$$$-by-$$$2$$$, $$$10$$$-by-$$$5$$$, $$$3$$$-by-$$$3$$$ and $$$7$$$-by-$$$3$$$. Giving Alice the $$$10$$$-by-$$$5$$$ and $$$7$$$-by-$$$3$$$ pieces, she gets a total of $$$50 + 21 = 71$$$ chocolate cells.